Simplify the following expression and state the conditions under which the simplification is valid. You can assume that $y \neq 0$. $x = \dfrac{-5y + 15}{y^2 - 13y + 30} \times \dfrac{2y - 20}{2y + 4} $
Solution: First factor the quadratic. $x = \dfrac{-5y + 15}{(y - 10)(y - 3)} \times \dfrac{2y - 20}{2y + 4} $ Then factor out any other terms. $x = \dfrac{-5(y - 3)}{(y - 10)(y - 3)} \times \dfrac{2(y - 10)}{2(y + 2)} $ Then multiply the two numerators and multiply the two denominators. $x = \dfrac{ -5(y - 3) \times 2(y - 10) } { (y - 10)(y - 3) \times 2(y + 2) } $ $x = \dfrac{ -10(y - 3)(y - 10)}{ 2(y - 10)(y - 3)(y + 2)} $ Notice that $(y - 3)$ and $(y - 10)$ appear in both the numerator and denominator so we can cancel them. $x = \dfrac{ -10(y - 3)\cancel{(y - 10)}}{ 2\cancel{(y - 10)}(y - 3)(y + 2)} $ We are dividing by $y - 10$ , so $y - 10 \neq 0$ Therefore, $y \neq 10$ $x = \dfrac{ -10\cancel{(y - 3)}\cancel{(y - 10)}}{ 2\cancel{(y - 10)}\cancel{(y - 3)}(y + 2)} $ We are dividing by $y - 3$ , so $y - 3 \neq 0$ Therefore, $y \neq 3$ $x = \dfrac{-10}{2(y + 2)} $ $x = \dfrac{-5}{y + 2} ; \space y \neq 10 ; \space y \neq 3 $